Integration – Overall Objectives  Integration as the inverse of differentiation  Definite and indefinite integrals  Area under the curve

Integration  Differentiation and Integration are inverses  Multiply by power and reduce power by 1  Add 1 to power and divide by new power This is called an INDEFINITE INTEGRAL, since you don’t know the constant (c)

Area under a straight line y = 0.5x + 3 What is the area under the line between x=1 and x=4? When x = 4 y = 0.5 x = 5 5 When x = 1 y = 0.5 x = Area triangle = 0.5 x (4-1) x (5-3.5) = 0.5 x 3 x 1.5 = 2.25 Area rectangle = (4-1) x 3.5 = 3 x 3.5 = 10.5 TOTAL AREA = = 12.75

Area under a curve 1 4 y = 0.5x What is the area under the curve between x=1 and x=4? Can’t do it by triangles and rectangles

The definite Integral  Area under a curve between x=a and x = b is given by the definite integral  Areas below the x axis are negative

The definite Integral  The definite integral a and b represent the limits of integration … these are the two values the integral is evaluated between This is called a DEFINITE INTEGRAL, since it can be evaluated to a value

The definite Integral – example 1  Evaluate Integrate The limits are written outside a square bracket Note: no constant of integration is included The definite integral is evaluated by “f(2) - f(0)” Substitute…

The definite Integral - example 2  Evaluate Integrate The limits are written outside a square bracket Note: no constant of integration is included The definite integral is evaluated by “f(3) - f(1)” Substitute…

The definite Integral - example 3  Evaluate Integrate Substitute… -2 2

Rules for Definite Integrals